Symmetry and uniqueness via a variational approach

Abstract: For some nonlocal PDEs, their steady states can be seen as critical points of some associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progress in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan). I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

analysis of PDEs

Audience: researchers in the topic

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Seminar In the Analysis and Methods of PDE (SIAM PDE)

Series comments: The Editorial Board of SIMA and the officers of the SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) are launching the Seminar In the Analysis and Methods of PDE (SIAM PDE) to be held on the first Thursday of the month at 11:30am ET, except in August and January. The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

This webinar is intended to continue well beyond this social distancing period. There are two type of SIAM PDE webinars: (i) Colloquium-type webinars by outstanding speakers presenting a survey of results on a topic where critical progress was recently achieved (and not necessarily published in SIMA); (ii) Webinars by outstanding authors based on some of the best articles accepted by SIMA. To launch the webinar series the initial six seminars will be of the Colloquium-type.

The talks are open to the public and, due to security reasons, all attendees have to register by following the link siam.zoom.us/webinar/register/WN_wShvDipBQw6WafNN6QO5wg

The link will contain detailed information about the talks. The registration is quick (asks only for your name and email) and, once registered, you will receive an email with the link to the webinars, which is unique to you, so please do not share that email. The registration is valid for all this webinar series, and you will receive a reminder about future talks.

Further, you will receive a follow-up email up to 7 days after each webinar, with a link to the on-demand SIAM PDE YouTube playlist recording.

Updated information and schedule will be available at sinews.siam.org/Details-Page/seminar-in-the-analysis-and-methods-of-pde

The SIAM PDE Webinar was initiated on June 2020:

By the SIMA Editors:

Robert Lipton, SIMA Editor-in-Chief (Louisiana State University, USA)

Athanasios Tzavaras (KAUST, Saudi Arabia)

By the SIAG/APDE Officers:

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